MATH 25 Notes 2015

§1 - Review

Solving Equations

Example 0.1. 3y + 2[5(y − 4)− 2] = 5y + 6(7 + y)− 3

Example 0.2. (2y − 3)13 − (4y + 5)

13 = 0

Solving for a variable

Example 0.3. a2 + b2 = c2 Solve for a

Example 0.4. A = P + Prt Solve for r

Example 0.5. A = πr2 Solve for r

Solving Inequalities

Example 0.6. −2(x+ 3) < 10 Solve for x in interval notation

Example 0.7. 4x− 1 ≤ 3x− 4 Solve for x in interval notation

Example 0.8. −3 ≤ 2x+ 5 < 17 Solve for x in interval notation

Example 0.9. |x− 13|+ 4 ≤ 5 Solve for x in interval notation

Word Problem

Example 0.10. How much 80% antifreeze solution should be mixed with 2 gallons of 50% antifreeze solutionto make a 60% antifreeze solution?

§2.4 - Linear Equations in Two Variables

There are three forms for writing the equation of a line. They are as follows:A linear equation can be written in standard form as: ax+ by = cA linear equation can be written in point-slope form as: y − y1 = m(x− x1)A linear equation can be written in slope-intercept form as: y = mx+ b

Example 0.11. Graph the following lines:

2x+ 3y = 6

x = 3

y = 2

Example 0.12. Find the slope of the three lines in the above problem.

Example 0.13. Find the slope of the line passing through the points (−3,−2) and (2, 5).

Example 0.14. Graph a line with slope −4 and passes through the point (2,−3).

The average rate of change of a function, f(x) between two points, (x1, f(x1)) and (x2, f(x2)) is given bythe following equation:

Average Rate of Change =f(x2)− f(x1)

x2 − x11

2

Example 0.15. Find the average rate of change of the function f(x) = x2 − 1 between the points x1 = −2and x2 = 0.

Example 0.16. Solve the following equations and inequalities by graphing:

2x− 3 = x− 1

2x− 3 < x− 1

2x− 3 > x− 1

Example 0.17. Solve 6x− 2(x+ 2)− 5 ≤ 0 by graphing.

§2.5 - Applications of Linear Equations

Example 0.18. Using the point slope formula, write the equation of the line passing through (4,−6) and(1,−2).

If m1 and m2 are slopes of 2 non-vertical parallel lines, then m1 = m2.If m1 and m2 are slopes of 2 non-vertical perpendicular lines, then m1 = − 1

m2or m1m2 = −1.

Example 0.19. For each of the following slopes m1, write the slope m2 that is perpendicular to it:

m1 = 2

m1 = −3

m1 = −1

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Example 0.20. Write an equation of a line passing through (−4, 1) and parallel to the line x+ 4y = 3.

Example 0.21. Write an equation of a line passing through (−3, 2) and parallel to the line x+ 3y = 6.

Example 0.22. Write an equation of a line passing through (2,−3) and perpendicular to the line y = 12x−4.

Example 0.23. Write an equation of a line passing through (−8,−4) and perpendicular to 6y − x = 18.

Linear Cost Function: C(x) = mx+ bRevenue Function: R(x) = pxProfit Function: P (x) = R(x)− C(x)

Where m is the variable cost, b is the fixed cost, x is the number of items, and p is the selling price.

Example 0.24. A family phone plan has a monthly base price of $99 plus $12.99 for each additional phoneadded to the plan. Write the linear cost function C(x). Compute C(4). What does this answer mean?

Example 0.25. An art show vendor sells lemonade for $2.00 per cup. The cost to rent the booth at theshow costs $120. Supplies to make and serve lemonade cost $.50 per cup of lemonade.a) Write the cost function.b) Write the revenue function.c) Write the profit function.d) How much profit is made if 50 cups are sold?e) How much profit is made if 128 cups are sold?f) How many cups must be sold for the vendor to break even?

Example 0.26. If the fixed cost of a business is $2275, its variable cost is $34.50, and the price it sells eachitem is $80, then write the three functions C(x), R(x), and P (x).

Example 0.27. Jorge borrows $2400 from his grandmother and pays the money back at a rate of $150 permonth.a) Write the linear function L(x) for the amount of money that Jorge still owes his grandmother at x months.b) Calculate L(12) and explain its meaning.

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Example 0.28. A car has a 15 gallon tank for gas and gets 30 miles to the gallon on a highway whendriving 60 miles per hour. If he starts with a full tank of gas (15 gallons), and travels 450 miles at 60 milesper hour, thena) Write the G(t) function for the amount left in the tank.b) Find G(4.5) and explain its meaning.

Example 0.29. A dance studio has a fixed cost of $1500. The studio charges $60 for each lesson. Thestudio has a variable cost of $35 to pay instructors.a) Write the cost function.b) Write the revenue function.c) Write the profit function.d) Determine the number of lessons to make a profit.e) If 82 lessons are given in one month how much does the studio make?

§2.6 - Transformations

Below is a chart that summarizes all of the transformations we covered in class. The general formula isin the second column, and there is an example given for each case for the function f(x) = |x| (the absolutevalue function). You can use the second column as a guide to work with any function that you are given. Inclass we will work with f(x) = |x|, f(x) = x2, and f(x) = x3.

Figure 1. Transformation Chart - From http://hellermaayanotmath.wikispaces.com

Below are a few examples of using transformations that may be helpful for studying.

Example 0.30. Apply a horizontal translation 3 units left and a vertical translation of 4 units down for thegraph of f(x) = x2, and write the formula for this graph.

Example 0.31. Apply a horizontal translation 2 units right, a vertical translation of 5 units up, and areflection over the x-axis for the graph of f(x) = |x|, and write the formula for this graph.

Example 0.32. Sketch the graph of f(x) = (x+ 5)5 − 2 using graph transformations.

Example 0.33. Sketch the graph of f(x) = (x+ 1)3 + 2 using graph transformations.

Example 0.34. Sketch the graph of f(x) = −|x− 3|+ 4 using graph transformations.

Example 0.35. Sketch the graph of f(x) = −2(x+ 1)2 − 5 using graph transformations.

Example 0.36. Sketch the graph of f(x) =√x+ 5 using graph transformations.

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§2.7 - Graphs of Functions and Piecewise Functions

Symmetry Test:1) The graph of a function is symmetric about the y-axis if we substitute −x for x, and reduce, then we getthe same function we started with.2) The graph of a function is symmetric about the x-axis if we substitute −y for y, and reduce, then we getthe same function we started with.3) The graph of a function is symmetric about the origin if we substitute −x for x AND −y for y, andreduce, then we get the same function we started with.

Check the above 3 conditions on the following functions to check for symmetry:

Example 0.37. y = |x|

Example 0.38. x = y2 − 4

Example 0.39. y = x2

Example 0.40. y = x3

Even and Odd Test:1) The graph of a function is even if f(−x) = f(x) for all x in the domain of the function. The graph of aneven function is symmetric about the y-axis.2) The graph of a function is odd if f(−x) = −f(x) for all x in the domain of the function. The graph of anodd function is symmetric about the origin3) If the function f(x) does not satisfy either condition above, the function is neither even nor odd.

Check the above 3 conditions on the following functions to check if the function is even, odd, or neither:

Example 0.41. f(x) = −2x4 + 5|x|

Example 0.42. f(x) = 4x3 − x

Example 0.43. f(x) = −x5 + x3

Example 0.44. f(x) = x2 − |x|+ 1

Example 0.45. f(x) = 2|x|+ x

Example 0.46. f(x) = x4 + x2 + x+ 1

Piecewise Functions

Evaluate the following function at the given points:

f(x) =

−x− 1 for − 4 ≤ x < −1−3 for − 1 ≤ x < 2√x− 2 for x ≥ 2

a) f(−3)b) f(−1)c) f(2)d) f(6)

Graphing Piecewise Functions

Here are some guided steps that you can use to always graph piecewise functions accurately:1) Draw your x and y axes (if none are provided for you).2) Plot your points that are at end of the intervals first (the ones that are given next to each piece (ex.−4,−1, 2 in the previous example). Be sure to pay attention whether the dot must be closed or open.3) Draw lightly a vertical dotted line through each point. This will let you know that each segment of thegraph can only be located in the regions that you have sliced the plane into.4) Draw the graph the question indicates in each region.5) The graph should be completed in each section, this is your piecewise graph.

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Here are some examples to try. Graph the following piecewise functions:

Example 0.47.

f(x) =

{x+ 3 for x < −1x2 for − 1 ≤ x < 2

Example 0.48.

f(x) =

{|x| for − 4 ≤ x < 2x2 for x ≥ 2

Example 0.49.

f(x) =

{−x+ 1 for x < 1√x for 1 ≤ x < 4

Example 0.50.

f(x) =

{x2 + 1 for x < −22x+ 3 for x ≥ 2

Example 0.51.

f(x) =

{2 for x ≤ −1−2x for x > −1

Example 0.52.

f(x) =

3x+ 3 for x < 1x2 for 1 ≤ x < 2−x− 1 for x ≥ 2

Greatest Integer Function

The greatest integer function (or floor function) is a special piecewise defined graph. it is defined as

f(x) = [[x]] where [[x]] denotes the greatest integer less than or equal to x

Evaluate the greatest integer function at the given points:a) f(1)b) f(1.7)c) f(−2.2)d) f(3.5)

Increasing, Decreasing, and Constant Functions

Suppose that I is an interval contained within the domain of f(x)

1) f(x) is an increasing on I if f(x1) < f(x2) for all x1 < x2 in I.2) f(x) is an decreasing on I if f(x1) > f(x2) for all x1 < x2 in I.3) f(x) is an increasing on I if f(x1) = f(x2) for all x1 and x2 in I.

Find where the following functions are increasing, decreasing, or constant:

Example 0.53. f(x) = x2

Example 0.54. f(x) = |x+ 1|

Example 0.55. f(x) = −2x+ 1

Relative Minimum and Maximum Values

Let f(x) be a function, and x = a and x = b be two points. Then f(a) is a relative maximum of f(x) ifthere exists an open interval containing a such that f(a) ≥ f(x) for all x in the interval. Alternatively, f(b)is a relative minimum of f(x) if there exists an open interval containing b such that f(b) ≤ f(x) for all x inthe interval. An open interval is an interval that does not include the endpoints.

Find where the following functions have relative maxima and minima:

Example 0.56. f(x) = x2

Example 0.57. f(x) = |x+ 1|

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Example 0.58. f(x) = −(x− 2)2 + 1

§3.1 - Quadratic Functions

Definition 0.1. A quadratic function is of the form

f(x) = ax2 + bx+ c

Definition 0.2. The maximum or minimum of the parabola is called a vertex.

Definition 0.3. The vertical line that passes through the vertex is called the axis of symmetry.

The following proof is to show how to get the vertex form of the parabola in general. We will have tocomplete the square.

f(x) = ax2 + bx+ c

= a

(x2 +

b

ax

)+ c

= a

(x2 +

b

ax+

b2

4a2

)+ c− a

(b2

4a2

)= a

(x+

b

2a

)2

+ c− b2

4a

= a

(x+

b

2a

)2

+4ac− b2

4a

= a(x− h)2 + k

where we define h = − b2a and k = 4ac−b2

4a . So then we have that

Vertex =

(− b

2a, f

(− b

2a

))Axis of Symmetry⇒ x = − b

2a

Definition 0.4. If we have a quadratic function f(x) = ax2 + bx+ c, then(i) a > 0 implies that the vertex is a minimum.(ii) a < 0 implies that the vertex is a maximum.

Definition 0.5. If we have a quadratic function f(x) = ax2 + bx+ c, then(i) If b2 − 4ac > 0⇒, then f has 2 real roots.(ii) If b2 − 4ac = 0⇒, then f has 1 real root.(iii) If b2 − 4ac < 0⇒, then f has no real roots (does not cross the x-axis).

Below are a few examples of writing the quadratic in vertex form.

Example 0.59. Rewrite the function f(x) = x2 − 8x+ 7 in vertex form.

Example 0.60. Rewrite the function f(x) = x2 + 6x+ 1 in vertex form.

Example 0.61. Rewrite the function f(x) = 3x2 + 12x+ 2 in vertex form.

Example 0.62. Rewrite the function f(x) = 4x2 − 40x+ 13 in vertex form.

§3.2 - Introduction to Polynomials

Definition 0.6. A polynomial function is of the form

f(x) = anxn + an−1x

n−1 + ...+ a1x+ a0

Definition 0.7. The degree of a polynomial is the highest value exponent of the function f(x) above.

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Here is a good chart to summarize the end behavior of polynomials that we covered in class. This includesall the necessary information (this is also called the leading term test in the textbook) to know along withsome examples of each.

Figure 2. End Behavior Chart - From http://www.shelovesmath.com/algebra/advancedalgebra/graphing-polynomials/

Definition 0.8. The multiplicity of f(x) = (x − a)n is the power n. If n is odd, then the graph crossesat the x-intercept (a, 0). If n is even, then the graph touches at the x-intercept (a, 0).

Theorem 0.1. Intermediate Value Theorem: Let f be a polynomial function. For a < b, if f(a) and f(b)have opposite signs, then f has at least one zero on the interval [a, b].

Example 0.63. Sketch the graph of f(x) = x(x+ 1)2(x− 1)2.

Example 0.64. Sketch the graph of f(x) = (x+ 5)4(x+ 1)3(x− 1)2.

Example 0.65. Sketch the graph of f(x) = −(x+ 1)2(x− 4)2.

Example 0.66. Sketch the graph of f(x) = −x(x+ 3)4(x− 3)2.

Example 0.67. Sketch the graph of f(x) = x4 − 2x2.

Example 0.68. Sketch the graph of f(x) = x3 − 9x.

Example 0.69. Sketch the graph of f(x) = 9x5 + 9x4 − 25x3 − 25x2.

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§3.3 - Polynomial Division and Factoring Theorems

Definition 0.9. We can write a polynomial f(x) as the following if d(x) 6= 0 and the degree of d(x) is lessthan or equal to the degree of f(x):

f(x) = d(x)q(x) + r(x)

Where f(x) is the dividend, d(x) is the divisor, q(x) is the quotient, and r(x) is the remainder.

Example 0.70. Use long division to divide: (6x3 − 5x2 − 3)÷ (3x+ 2).

Example 0.71. Use long division to divide: (3x4 + 2x3 + 4x2 + x− 5)÷ (3x+ 2).

Example 0.72. Use long division to divide: (4x3 − 23x+ 3)÷ (2x− 5).

Example 0.73. Use long division to divide: (2x4 − 3x3 + 5x2 − 7x+ 1)÷ (x2 + 3).

Example 0.74. Use long division to divide: (2x2 + 3x− 14)÷ (x− 2).

Example 0.75. Use long division to divide: (3x2 − 14x+ 15)÷ (x− 3).

Example 0.76. Use synthetic division to divide: (3x2 − 14x+ 15)÷ (x− 3).

Example 0.77. Use synthetic division to divide: (−10x2 + 2x3 − 5)÷ (x− 4).

Example 0.78. Use synthetic division to divide: (4x3 − 28x− 7)÷ (x− 3).

Example 0.79. Use synthetic division to divide: (x4 + 4x3 − 2x+ 18)÷ (x+ 3).

Example 0.80. Use synthetic division to divide: (2x4 + 7x3 − 3x+ 5)÷ (x+ 1).

Example 0.81. Given f(x) = x4 + 6x3 − 12x2 − 30x+ 35, use synthetic division and the remainder to findf(2) and f(−7).

Example 0.82. Given f(x) = x4 + x3 − 6x2 − 5x − 15, use synthetic division and the remainder to findf(5) and f(−3).

Example 0.83. Given f(x) = 2x4 − 4x2 − 13x− 9, use synthetic division and the remainder to determineif c = 4 is a zero of f(x).

Example 0.84. Given f(x) = x3 + x2 − 3x − 3, use synthetic division and the remainder to determine if

c =√

3 is a zero of f(x).

Example 0.85. Given f(x) = x3+x+10, use synthetic division and the remainder to determine if c = 1+2iis a zero of f(x).

Example 0.86. Given f(x) = x4 − x3 − 11x2 + 11x+ 12, use synthetic division and the factor theorem todetermine if (x− 3) and (x+ 2) are factors f(x).

Example 0.87. Given f(x) = 2x4 − 13x3 + 10x2 − 25x + 6, use synthetic division and the factor theoremto determine if (x− 6) and (x+ 3) are factors f(x).

Example 0.88. Factor f(x) = 3x3 + 25x2 + 42x − 40, given that −5 is a zero of f(x). Then solve theequation 3x3 + 25x2 + 42x− 40 = 0.

Example 0.89. Factor f(x) = 2x3+7x2−14x−40, given that −4 is a zero of f(x). Then solve the equation2x3 + 7x2 − 14x− 40 = 0.

Example 0.90. Write a polynomial f(x) of degree three that has roots x = 1, x = 2, and x = 3.

Example 0.91. Write a polynomial f(x) of degree three that has roots x = 13 , x =

√6, and x = −

√6.

Example 0.92. Write a polynomial f(x) of degree three that has roots x = 2, x = 1 + i, and x = 1− i.